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arXiv:2407.03206 · Physical review. B./Physical review. B

Multipartite Greenberger-Horne-Zeilinger entanglement in monitored random Clifford circuits

Guanglei Xu, Yu-Xiang Zhang

Published Nov 24, 2025• 2 citations
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Abstract

Interactions in many-body systems are typically short-range and few-body. We investigate how such local interactions build up long-range and intrinsically multipartite entanglement by studying the $n$-partite Greenberger-Horne-Zeilinger $({\text{GHZ}}_{n})$ entanglement in monitored random Clifford circuits, which is well-known for a measurement-induced transition between phases of volume-law and area-law (bipartite) entanglement. We obtain a series of results: (1) About 1.25 $|{\text{GHZ}}_{3}\ensuremath{\rangle}$ can be extracted from states in the volume-law phase. This value is remarkably universal, independent of both the measurement rate and partitioning details, until a phase transition (either measurement-induced or a newly identified partitioning-induced transition) is approached. (2) Dynamically, the creation (sometimes also the annihilation) of ${\text{GHZ}}_{3}$ entanglement occurs suddenly via dynamical phase transitions (DPTs). The critical points of these DPTs are governed by the entanglement speed $({v}_{E})$ of biaprtite entanglement. (3) In stark contrast to ${\text{GHZ}}_{n\ensuremath{\le}3}$, ${\text{GHZ}}_{n\ensuremath{\ge}4}$ entanglement is statistically significant only at the measurement-induced critical point, not in the bulk of the volume-law phase. Our results uncover a rich hierarchy of multipartite entanglement structures.

DOI
10.1103/mh1s-kbjl

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In many-body physics, an explicit wavefunction often provides the most thorough understanding, yet for monitored random Clifford circuit it has remained missing. Here we develop a framework that grants direct access to the typical output states. As every stabilizer state is local-Clifford-equivalent to a graph state, the graph adjacency matrix, a classical bit matrix, provides a complete description of the quantum state. In the large-NNN limit, we show that these graphs converge to the Erdős--Rényi random graph G(N,1/2)G(N,1/2)G(N,1/2), which allows us to resolve the open problem of Greenberger--Horne--Zeilinger (GHZ) entanglement generated by deep random Clifford circuits. We obtain analytically the mean GHZ content ⟨g3⟩=1.204\langle g_3\rangle=1.204⟨g3​⟩=1.204 for even NNN and 1.3251.3251.325 for odd NNN. For monitored Clifford circuits with a 1D layout, we uncover an emergent Erdős--Rényi subgraph G(Nsub,1/2)G(N_{\mathrm{sub}},1/2)G(Nsub​,1/2) in the output states of the volume-law phase, where Nsub/N≈1−p/pcN_{\mathrm{sub}}/N\approx \sqrt{1-p/p_c}Nsub​/N≈1−p/pc​​ with ppp the measurement rate and pcp_cpc​ the critical point of the measurement-induced phase transition (MIPT). The output state is thus equivalent to the output of an unmonitored random Clifford circuit on NsubN_{\mathrm{sub}}Nsub​ qubits, weakly perturbed by the remaining N−NsubN-N_{\mathrm{sub}}N−Nsub​ qubits carrying little entanglement. This result directly accounts for the quantum error-correcting capability of the volume-law phase, and implies the same GHZ statistics for the whole volume-law phase. We further identify a clustering effect for qubits in the dense subgraph, which we reproduce with an infection-recovery toy model that exhibits a measurement-induced absorbing-state phase transition. Finally, a mean-field argument on the graph locates the MIPT critical point at pc=0.1608p_c = 0.1608pc​=0.1608, in excellent agreement with the numerical value pc≈0.16p_c\approx 0.16pc​≈0.16.
We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finite-range two-body gates, or are pair-wise interactions randomly distributed throughout the system with a power-law distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurement-induced transitions. For the cluster unitary gates we find a transition between a phase with volume-law scaling of the entanglement entropy and a phase with area-law entanglement entropy. Our results indicate that the universality class of the phase transition is compatible to that of short range hybrid Clifford circuits. Oppositely, in the case of power-law distributed gates, we find the universality class of the phase transition changes continuously with the parameter controlling the range of interactions. In particular, for intermediate values of the control parameter, we find a non-conformal critical line which separates a phase with volume-law scaling of the entanglement entropy from one with sub-extensive scaling. Within this region, we find the entanglement entropy and the logarithmic negativity present a cross-over from a phase with algebraic growth of entanglement with system size, and an area-law phase.
Measurement-only circuits provide a minimal setting in which repeated local projections can either generate or suppress many-body entanglement, giving rise to measurement-induced phase transitions and dynamical regimes, that might have no unitary counterpart. Here we investigate entanglement and information transitions in one-dimensional measurement-only Clifford circuits with long-range two-qubit parity checks. By tuning both the measurement range and density per layer, we uncover a broad set of phases whose classification requires probes beyond entanglement entropy, such as mutual information, tripartite mutual information, purification from an ancilla, and Bell-cluster statistics. We map phase diagrams using large-scale Clifford simulations for two protocols: a random-basis design in which each measurement is randomly chosen from {XX,YY,ZZ}\lbrace XX,YY,ZZ \rbrace{XX,YY,ZZ}, and a single-basis design in which the basis is fixed within each layer but varies between layers, hence introducing more structure to the circuit. We map the trajectory-averaged entanglement entropy to a two-dimensional statistical mechanics model by extending a replica-based method to random-basis measurement-only circuits, and show that a continuous-time limit yields an effective long-range XX hamiltonian in the steady state. This connection links the observed volume-law to sub-volume-law entanglement transition to the boundary between a continuous symmetry broken phase and a critical XY phase. Strikingly, in structured (single-basis) circuits we find a phase in which volume-law and long-range entanglement coexists with rapid, size-independent purification of an ancilla qubit, and the absence of scrambling, highlighting measurement-only circuits as a promising route to efficiently preparing highly entangled and technologically useful quantum states.
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.
We study the structure of multipartite entanglement in monitored quantum circuits exhibiting measurement-induced phase transitions (MIPTs). Using a one-dimensional Clifford circuit subject to local measurements with a probability ppp, we show numerically that the entanglement depth, corresponding to the size of the largest cluster of entangled qubits scales as a power law with system size on both sides of the transition. The power law exponent is 1 in the entangling phase and continuously decreases to 0 as p→1p \to 1p→1 in the disentangling phase. In addition, we find that the spatial support of the largest cluster exhibits an approximate fractal geometry with a tunable fractal dimension controlled by the measurement rate. We argue that this structure arises from a competition between unitary-driven coagulation of entangled clusters and measurement-induced fragmentation, giving rise to a fractal steady state reminiscent of classical coagulation-fragmentation models. Away from the MIPT critical point, the fractal dimension matches the entanglement depth power law exponent. These results show that multipartite entanglement structure provides a fresh perspective on the emergent quantum correlations in monitored quantum circuits and noisy quantum dynamics.
An entanglement transition can occur without a corresponding transition in purification. We demonstrate this separation in spatially local monitored Clifford circuits with a fixed number of layers per Floquet period, where a spatially random pattern of Clifford gates and Pauli measurements is repeated exactly in time. Rare spatial structures persist under this repetition and break the correspondence between entanglement and purification found in conventional spacetime-random circuits. In one spatial dimension, the late-time entanglement produced with an initial product state remains area-law, while the system remains extensively mixed in the thermodynamic and long-time limits if the initial state is maximally mixed. In two or more dimensions, we find a transition between volume-law and area-law entanglement while the system remains mixed on both sides of this transition. An extensive amount of quantum information survives indefinitely in both phases and evolves unitarily despite the repeated measurements. We introduce the logical-support radius to characterize its spatial spreading and identify the volume-law and area-law phases with delocalized and localized logical dynamics, respectively. We further introduce the logical Krylov dimension, which counts the independent logical operators dynamically generated from an initially local operator. Its scaling resolves the transition and provides evidence for a new universality class of measurement-induced phase transitions.